\chapter{Test Problems and Further Research}

\section{Test problem for time discretization schemes}

To assess the stability of various implicit time integration procedures, we shall consider a simplified problem in one dimension given below as test problem.

\begin{subequations}
\begin{equation}
\frac{\partial \zeta}{\partial t} + \nabla .(\zeta\textbf{U}+h\nabla \varphi ) = 0,
\end{equation}
\begin{equation}
\frac{\partial \varphi}{\partial t} + \textbf{U}.\nabla \varphi + g\zeta = 0,
\end{equation}
\end{subequations}

Here, we have neglected the impact of the vertical structure $\psi$ and pressure pulse $P_s$. This is now a coupled initial boundary value problem, where both equations are hyperbolic in nature and also represents an Hamiltonian Set. Also, the boundary conditions for both $\zeta$ and $\varphi$ are taken as periodic.


Assuming no spatial variation of mean current velocity $U$ and depth $h$, the above equations can be written as:


\begin{subequations}
\begin{equation}
\frac{\partial \zeta}{\partial t} +\textbf{U} \dfrac{\partial \zeta}{\partial x} + h \dfrac{\partial^2 \varphi}{\partial x^2} = 0,
\end{equation}
\begin{equation}
\frac{\partial \varphi}{\partial t} + \textbf{U}\dfrac{\partial \varphi}{\partial x} + g\zeta = 0,
\end{equation}
\end{subequations}

Implicit time integration procedures will require solving a system of linear equations. We will use MATLAB to solve the system after performing spatial discretization and applying the boundary conditions. The idea here is to assess the stability and accuracy of various methods and not the performance.

\section{Test problem for generalized Krylov Subspace methods}
In order to validate and analyze the performance of the generalized Krylov subspace methods, we will test the method with Poisson equation given by:
\begin{subequations}
\begin{align}
 -\Delta u &=f(x,y) \text{\quad on \quad} \Omega = (0,1) \times (0,1)\\
 u (x,y ) &= 0 \text{\qquad \quad on \quad} \delta \Omega.
\end{align}
\end{subequations}

The discretization of the Poisson equation results in a symmetric matrix where both CG and generalized Krylov Subspace methods can be applied.  The RRB-k PCG method developed in \cite{Jong} has been tested for the Poisson equation. We will implement the generalized Krylov subspace methods and compare the performance and the storage requirements with that of RRB-k PCG method for the Poisson equation .


\section{Realistic problems {IJssel, Plymouth}}

Two realistic problems are obtained from MARIN's database and previous simulations by Martijn in \cite{Jong}. Testing will be carried out with respect to the validated models in \cite{Jong}. A very small time step (much lower than the allowable limit) will be used to first carry out simulations with the current model which use Explicit Leap Frog time integration. Results from implicit time integration will then be compared with the results from explicit time integration.

\subsection{The Gelderse IJssel}

The Gelderse IJssel, a small river, is a branch from the Rhine in the Dutch provinces Gelderland and Overijssel. The river flows from Westervoort and discharges in the IJsselmeer. In Figure 6.1 a (small) part of the river is shown. From this part several test problems are extracted. This is done by extracting small regions out of the the displayed region . For the discretization an equidistant 2 m by 2 m grid is used. During the later part of the thesis, non-uniform grids will be explored and the discretization modified accordingly.

\begin{figure}[H]
\label{Ijssel}
\centering
\includegraphics[width=0.6\textwidth]{ijssel}~\\[0.5cm]
\caption{The Gelderse IJssel (Google Maps)}
\end{figure}


\subsection{The Plymouth Sound}

Plymouth Sound is a bay located at Plymouth, a town in the South shore region of England, United Kingdom. The Plymouth Breakwater is a dam in the centre of the Plymouth Sound which protects anchored ships in the Northern part of the Plymouth Sound against south-western storms. From this region also test problems are extracted, see Figure 6.2. For the discretization an equidistant 5 m by 5 m grid is used. For the non-uniform grid structure, discretization will be modified accordingly.

\begin{figure}[H]
\label{plymouth}
\centering
\includegraphics[width=0.6\textwidth]{plymouth}~\\[0.5cm]
\caption{The Plymouth Sound (Google Maps)}
\end{figure}

\newpage
\subsection{Research Questions}

Following research questions and goals, are the topics that will be treated during the rest of the research:

\begin{itemize}
 \item Implement generalized Kyrlov subspace method.
  \begin{itemize}
  \item Analyze the impact on the performance (speed-up) and storage requirements for the generalized Kyrlov Subspace methods for CUDA and C++ code.
   \end{itemize}
 \item Implement various time integration schemes.
 \begin{itemize}
  \item Analyze the stability and accuracy of various time integration schemes and assess the storage and performance requirements.
   \item From the current time step of 0.01 seconds, how much improvement without loss of accuracy and performance can be achieved.
 \end{itemize}
  \item If time integration schemes do not yield significant improvement in stability, analyze the possibility of using the non-uniform mesh.
   \begin{itemize}
    \item Provide a framework for the use of either Adaptive mesh refinement or moving mesh method.
     \end{itemize}
\end{itemize}
